Conjugate Gradient Type Methods for Ill-Posed Problems

1. Auflage 1995
ISBN: 978-0-582-27370-2
Verlag: Taylor & Francis Ltd

Buch, Englisch, 144 Seiten, Format (B × H): 174 mm x 246 mm, Gewicht: 297 g

Reihe: Chapman & Hall/CRC Research Notes in Mathematics Series

The conjugate gradient method is a powerful tool for the iterative solution of self-adjoint operator equations in Hilbert space.This volume summarizes and extends the developments of the past decade concerning the applicability of the conjugate gradient method (and some of its variants) to ill posed problems and their regularization. Such problems occur in applications from almost all natural and technical sciences, including astronomical and geophysical imaging, signal analysis, computerized tomography, inverse heat transfer problems, and many more

This Research Note presents a unifying analysis of an entire family of conjugate gradient type methods. Most of the results are as yet unpublished, or obscured in the Russian literature. Beginning with the original results by Nemirovskii and others for minimal residual type methods, equally sharp convergence results are then derived with a different technique for the classical Hestenes-Stiefel algorithm. In the final chapter some of these results are extended to selfadjoint indefinite operator equations.

The main tool for the analysis is the connection of conjugate gradient
type methods to real orthogonal polynomials, and elementary
properties of these polynomials. These prerequisites are provided in
a first chapter. Applications to image reconstruction and inverse
heat transfer problems are pointed out, and exemplarily numerical
results are shown for these applications.

Zielgruppe

Professional Practice & Development

Autoren/Hrsg.

Hanke, Martin

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen

Inhaltsverzeichnis

1. Conjugate Gradient Type Methods 2. Regularizing Properties of MR and CGNE 3. Regularizing Properties of CG and CGME 4. On the Number of Iterations 5. A Minimal Residual Method for Indefinite Problems

Über Autor(innen)

Hanke\, Martin